The problem of allocating of multiple sensors for the surveillance of multiple targets advanced in radar systems are formidable. The sensors are often of different types and are usually located on different platforms while the surveillance is confined to a particular Volume of Interest (VOI). Target ranking, spatial and temporal coverage, conversion of assignments into sensor control commands, immunity to countermeasures, emission control, cueing and handoff, event prediction, and the ability of sensors to observe targets are among the many complicated aspects of the problem. Prior allocation methods were often ad hoc and sub-optimal.
The mathematical foundations of the invention are shown in equations 1-5 below. Let X be a discrete parameter space with a finite number of parameters. The most general way of describing any state of information on X is by defining a probability measure of .mu. that extends over X. The most familiar case is when .mu. is a probability for a given state of information and f(x) is its corresponding probability density function. Thus for any A X, where A is equal to, or is in, the set X, we have ##EQU1##
The probability density function f(x) is said to represent the state of information. Each track process is described by a Kalman filter. It is a property of the Kalman filter that one can, a priori, calculate the updated covariance matrix for the target, if it is assumed that a selected sensor will be used to observe or track the state vector. The form of the uncertainty of the components of the state vector x is selected as a gaussian multi-variate distribution. The form of the density function f(x; m,P) is then given by EQU .function.(x, m, P)=([2.pi.].sup.n/2 [det P].sup.1/2).sup.-1 exp (-1/2[x-m].sup.T P.sup.-1 [x-m]) (2)
where m is the mean vector of the state and P is the covariance matrix. The mean vector m is the most probable value of the state and the covariance matrix P is a symmetric matrix whose diagonal elements are the variances of the components of the state vector. In order to evaluate the efficacy of assigning a sensor to a target, we choose to use the predicted information content of the state vector as a comparative measure of how to make the best assignment.
The relative information content I(f,.lambda.) of a probability density function f(x) with respect to the (normalized) non-informative probability density .lambda.(x) is given by: EQU I(.function.,.lambda.)=.intg.d x.function.(x) ln (.function.(x)/.lambda.(x))(3)
and is simply called the information content of f(x). When the logarithm is to the base 2, the unit of information is termed a bit; if the base is e=2.71828 . . . , the unit is the nep or neper; and when the base is 10 the unit is the digit. It is known that I(f,.lambda.) is invariant under a change of variables that is one-to-one and onto (called bijective). It is also known that .lambda. is a uniform probability density if the coordinate system is Cartesian.
If sensors are assigned to observe the state vector x, then the new updated covariance matrix will be called P.sub.1, while the updated covariance matrix if the state vector is not observed will be called P.sub.o. The density function for the one step ahead (i.e., one calculation cycle) state vector if the track is observed by a sensor is f.sub.i (x;m.sub.1,P.sub.1). The density function for the one step ahead updated state vector if the track is not observed by a sensor is f.sub.o (x;m,P.sub.o). When the information gain in the one step ahead state vector for a track T is the difference in the information content if the track is observed minus the information content if the track is not observed. This "gain" in information can be denoted by G where EQU G=I(.function..sub.1,.lambda.)-I(.function..sub.o,.lambda.)(4)
The notation for the gain in assigning sensor i to track j is obtained by subscripting the gain as G.sub.ij. If the density functions are Gaussian, we can then evaluate I in closed form, and, therefore, also G to obtain for the gain in information content, the expression ##EQU2## where the notation .vertline.P.vertline. stands for the determinant of P and the unit is the neper.
The use of Kalman filters to obtain preferred sensor-to-target assignments in a generic surveillance context is known. Jeffrey M. Nash's article in Proceedings of the IEEE Conference on Decision and Control, 1997 entitled "Optimal Allocation of Tracking Resources" discusses an approach to the general problem. The Nash paper utilizes pseudo sensors in addition to the original independent sensors in which the pseudo sensors are provided for all possible sensor combinations. The Kalman filter is utilized to provide the covariance matrices P.sub.1 and P.sub.0 which are used in G to calculate the gains in information content which can then be presented in an augmented sensor target assignment array in which the targets 1-n are aligned in a row across the top of the array, and the sensors 1-m are aligned in a column on the left side of the array. Thus there will be formed n.times.m cells in the array, and a maximum information gain and associated sensor to target assignment will be calculated by a digital computer using a linear programming algorithm on a step-by-step basis. The assignment thus obtained is the optimal assignment in the sense of maximum gain of information.